Messi was awarded the Golden Ball as the best player of the tournament in Brazil, but his team slumped to a 1-0 defeat after extra time in the decider.
The Barcelona star was given his prize immediately after the game at the Maracana in Rio de Janeiro and Blatter said the 28-year-old acknowledged he had fallen short from a team perspective again.
"Messi was talking to himself," Blatter told Argentine publication La Nacion.
"He said to himself over and over again: 'The best but not the champion.'"
A five-time Ballon d'Or winner, Messi can add a major continental trophy to his collection with Argentina into the Copa America Centenario quarter-finals in the United States.